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Full text of "Treatise On Analysis Vol-Ii"

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11 THE SPACES L1 AND L2 165 sequence of functions consisting of 1 and /'" (n J> 1), where lim an = -f oo , is a total sequence in the space ^n(I), if and only if ]T a^1 = + co ("Muntz's theorem"). In -^R(X, p) or ^c(X, /*) a sequence (/) of functions is said to be orthonormal if the sequence of classes fn of these functions is an orthonormal system in the corre- sponding Hilbert space. (a) Let (fn) be an infinite orthonormal sequence, uniformly bounded on X. If p, is bounded, show that there exists a set A with measure >0 such that£ \fn(x)\2 = + oo n for all x e A. (Argue by contradiction, observing that the sequence (fn) cannot tend to 0 almost everywhere.) (b) Suppose that p, is bounded. Let (/) be a uniformly bounded infinite orthonormal sequence. If a sequence (an) of scalars is such that the series with general term anfn(x) converges almost everywhere, show that lim an = 0 (use EgorofTs theorem applied to H-+00 the sequence (\anfn\2)). Show that there exists a sequence of scalars (bn) such that lim bn = 0 and ]T I bn \ 2 \fn(x) \2 = + oo on a set of measure >0. (Choose the sequence n-*oo « (bn) so that ]£ | bn \ 2 = + oo , and argue by contradiction, by applying EgorofTs n theorem to the partial sums of the series ]T | bn \ 2 \fn \ 2.) n (c) Let (/) be a sequence in &2. Show that the following three conditions are equivalent : a) If a sequence (an) is such that the series with general term anfn(x) converges on a set of measure >0, then lim an = 0. n-+oo ft) If a sequence (an) is such that the series with general term anfn(x) converges absolutely on a set of measure >0, then lim an 0. n-*oo y) For every measurable set A of measure >0, we have liminf \fn\ dp>0. w-foo JA (To prove that y) implies a) and ft), use EgorofF s theorem. To prove the converse, argue by contradiction , by considering a subsequence (fnk) such that | fnk \ dp, > 1 Ik 3, and taking ank k and an 0 if n 9*= nk .) (d) Suppose that ^ is bounded and (/) is a sequence in 3? 2. Prove that the following three conditions are equivalent : a) For each sequence (an) such that the series with general term anfn(x) con- verges almost everywhere, we have lim an 0. n~+ oo /?) For each sequence (an) such that the series with general term anfn(x) converges absolutely almost everywhere, we have lim an 0. n-*oo y) There exists 8 > 0 such that, for every measurable set A whose complement has measure <|S, we have lim inf |/J dju. > 0. (Same method as (c).) Deduce that, H_KJO JA if the sequence (/) is orthonormal and uniformly bounded, it satisfies these three conditions, and therefore the series with general term aHfn(x) converges absolutely almost everywhere if and only if 2] |an| < + oo (use (b)).ontinuous affine linear functions on E is a subspace of the separable